A126853 Table read by antidiagonals: B(n,m) is the numerator of the Bernoulli polynomial of order m and degree n evaluated at x=0.

1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 0, 0, 5, -3, 1, 0, -1, -1, 2, -2, 1, 0, 0, 1, -9, 11, -5, 1, 0, 1, 1, 19, -6, 35, -3, 1, 0, 0, -5, -3, 251, -25, 17, -7, 1, 0, -1, -1, -16, -9, 24, -45, 35, -4, 1, 0, 0, 7, 5, 221, -475, 274, -147, 46, -9, 1, 0, 5, 3, 19, 11, 4315, -120, 1624, -56, 39, -5, 1, 0, 0, -15, -63, -199, -475, 19087, -294, 967, -81, 145, -11, 1

Offset

0,13

Comments

Absolute values of the diagonal are in A002657. Not to be confused with the poly-Bernoulli numbers.

Let H(k) = Sum_{i=1..k+1} 1/i. Then Seq((-1)^k*T(k,k+2),k=0..) = Seq(k!*H(k),k=0..) = 1,3/2,11/3,25/2,274/5,294,.. (Cf. A160039 and A014973). [Peter Luschny, Apr 30 2009]

Links

Table of n, a(n) for n=0..90.

D. Cvijovic and H. M. Srivastava, Closed form summation of the Dowker and related sums, J. Math. Phys. 48 (2007) 043507.

Index entries for sequences related to Bernoulli numbers.

Formula

E.g.f.: [t/(exp(t)-1)]^m*exp(t*x)=sum_{n=0..infinity} B_n^m(x)*t^n/n!.

Example

Table of fractions B(n,m) is read along antidiagonals and starts in row n=0 and column m=0:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, -1/2, -1, -3/2, -2, -5/2, -3, -7/2, ...
0, 1/6, 5/6, 2, 11/3, 35/6, 17/2, 35/3, ...
0, 0, -1/2, -9/4, -6, -25/2, -45/2, -147/4, ...
0, -1/30, 1/10, 19/10, 251/30, 24, 274/5, 1624/15, ...
0, 0, 1/6, -3/4, -9, -475/12, -120, -294, ...
0, 1/42, -5/42, -16/21, 221/42, 4315/84, 19087/84, 720, ...
0, 0, -1/6, 5/4, 11/3, -475/12, -1375/4, -36799/24, ...

Mathematica

B[n_, m_] := n! SeriesCoefficient[(t/(E^t-1))^m E^(t x), {t, 0, n}] /. x -> 0 // Numerator; Table[B[n-m, m], {n, 0, 12}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 21 2019 *)

Crossrefs

Sequence in context: A282887 A265729 A181886 * A286127 A201654 A265606

Adjacent sequences: A126850 A126851 A126852 * A126854 A126855 A126856

Keyword

frac,sign,tabl

Author

R. J. Mathar, May 13 2007, May 17 2007

Status

approved